We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results concerning the relations between these functions as well as their typical and extreme values.

In this article we prove finite generation of the cohomology of quotients of a PBW algebra A by relating it to the cohomology of quotients of a quantum symmetric algebra S which is isomorphic to the associated graded algebra of A. The proof uses a spectral sequence argument and a finite generation lemma adapted from Friedlander and Suslin.

In this note we consider a question of Ono, concerning which spaces of classical modular forms can be generated by sums of η-quotients. We give some new examples of spaces of modular forms which can be generated as sums of η-quotients, and show that we can write all modular forms of level Γ0(N) as rational functions of η-products.

This paper has two purposes. We determine the automorphism groups of the 2-transitive, bilinear dual hyperovals over F2 of type D[k], which were constructed in [6] by the author. Secondly, we characterize 2-transitive quotients of the Huybrechts dual hyperoval, compute their automorphism groups and give estimates on the number of such quotients.